TS-Reconfiguration of $k$-Path Vertex Covers in Caterpillars for $k \geq 4$

TL;DR

This paper presents a polynomial-time algorithm for the reconfiguration problem of $k$-path vertex covers under token sliding on caterpillar graphs for $k \\geq 4$, advancing understanding of its complexity on trees.

Contribution

It introduces the first polynomial-time solution for $k$-PVCR under token sliding on caterpillars for $k \\geq 4$, a step toward understanding the problem on trees.

Findings

Polynomial-time algorithm for caterpillars

Solves $k$-PVCR for $k \\geq 4$ on caterpillars

Progress toward complexity classification on trees

Abstract

A $k$-path vertex cover ($k$-PVC) of a graph $G$ is a vertex subset $I$ such that each path on $k$ vertices in $G$ contains at least one member of $I$. Imagine that a token is placed on each vertex of a $k$-PVC. Given two $k$-PVCs $I, J$ of a graph $G$, the $k$-Path Vertex Cover Reconfiguration ($k$-PVCR) under Token Sliding ($\mathsf{TS}$) problem asks if there is a sequence of $k$-PVCs between $I$ and $J$ where each intermediate member is obtained from its predecessor by sliding a token from some vertex to one of its unoccupied neighbors. This problem is known to be $\mathtt{PSPACE}$-complete even for planar graphs of maximum degree $3$ and bounded treewidth and can be solved in polynomial time for paths and cycles. Its complexity for trees remains unknown. In this paper, as a first step toward answering this question, for $k \geq 4$, we present a polynomial-time algorithm that solves $k$-PVCR under $\mathsf{TS}$ for caterpillars (i.e., trees formed by attaching leaves to a path).